Dirichlet Series with Periodic Coefficients and Their Value-Distribution near the Critical Line

Abstract: The class of Dirichlet series associated with a periodic arithmetical function f includes the Riemann zeta-function as well as Dirichlet L-functions to residue class characters. We study the value-distribution of these Dirichlet series L(s; f) and their analytic continuation in the neighbourhood of the critical line (which is the axis of symmetry of the related Riemann-type functional equation). In particular, for a fixed complex number a ≠ 0, we find for an even or odd periodic f the number of a-points of the Δ-factor of the functional equation, prove the existence of the mean of the values of L(s; f) taken at these points, show that the ordinates of these a-points are uniformly distributed modulo one and apply this to show a discrete universality theorem.